Editing Slash (punctuation) (section)

==== Quotient of set ====
{{see also|Set (mathematics)}}
A ''quotient of a set'' is informally a new set obtained by identifying some elements of the original set. This is denoted as a fraction <math>S / R</math> (sometimes even as a built fraction), where the numerator <math>S</math> is the original set (often equipped with some algebraic structure). What is appropriate as denominator depends on the context.

In the most general case, the denominator is an [[equivalence relation]] <math>\sim</math> on the original set <math>S</math>, and elements are to be identified in the quotient <math>S/{\sim}</math> if they are equivalent according to <math>\sim</math>; this is technically achieved by making <math>S/{\sim}</math> the set of all [[equivalence class]]es of <math>\sim</math>.

In [[group theory]], the slash is used to mark [[quotient group]]s. The general form is <math>G/N</math>, where <math>G</math> is the original group and <math>N</math> is the normal subgroup; this is read "<math>G</math> mod <math>N</math>", where "mod" is short for "[[modulo operation|modulo]]". Formally this is a special case of quotient by an equivalence relation, where <math>g \sim h</math> iff <math>g = hn</math> for some <math>n \in N</math>. Since many algebraic structures ([[Ring (mathematics)|ring]]s, [[vector space]]s, etc.) in particular are groups, the same style of quotients extend also to these, although the denominator may need to satisfy additional [[Closure (mathematics)|closure]] properties for the quotient to preserve the full algebraic structure of the original (e.g. for the quotient of a ring to be a ring, the denominator must be an [[Ideal (ring theory)|ideal]]).

When the original set is the set of [[integer]]s <math>\mathbb{Z}</math>, the denominator may alternatively be just an integer: <math>\mathbb{Z}/n</math>. This is an alternative notation for the set <math>\mathbb{Z}_n</math> of [[modular arithmetic#Integers modulo m|integers modulo ''n'']] (needed because <math>\mathbb{Z}_n</math> is also notation for the very different [[P-adic number|ring of ''n''-adic integers]]). <math>\mathbb{Z}/n</math> is an abbreviation of <math>\mathbb{Z}/n\mathbb{Z}</math> or <math>\mathbb{Z}/(n)</math>, which both are ways of writing the set in question as a quotient of groups.